Download The volume of the tetrahedron is

Inclusion of a regular tetrahedron in a cube
Teaching suggestions for a math laboratory activities with paper folding
Antonio Criscuolo Centro MatNet Università di Bergamo (Italy)
Francesco Decio CDO Bergamo (Italy)
Before starting a premise to be shared
This workshop is not intended as an activity to apply concepts and geometric properties, already
studied.
The workshop propose to the students:
• to be able to play their skills of observation, their spatial intuition, some
fundamental geometric skills of geometry of the space;
• for exploring geometric properties embedded in a geometric pattern origami
• to experiece simple spatial geometric questions and even discover unexpected
properties.
So, I think that the best way to present the workshop is creating
here the same atmosphere as in the class.
Therefore I ask you to perform the role of the
young students while I play the role of the old
teacher.
From a rectangular sheet to a first geometric origami
From a rectangular sheet to a first geometric origami
1
2
6
3
7
4
5
8
9
?
Francesco Decio’s diagrams
What kind of paper you need to build a tetrahedron origami?
•
•
sheet A4
Is it possible to construct a tetrahedron from a rectangular
sheet of any size?
For example: starting from an A4 paper sheet, can you
obtain equilateral triangles faces of the tetrahedron?
•
The rectangular sheet that we are using is a special sheet
......
•
... ... .... you can get the rectangular sheet cutting in half the
short side of an A4 sheet and then cut the strip that you has
obtained, so the long side is √3 times the short side (182mm
x 105 mm ).
How to make a 1: 3 sheet from an A4
Francesco Decio’s diagrams
From a A4 sheet to a six square sheets for the construction of a cube
If you fold the A4 sheet with one third of the long edge you will obtain
the six squares with maximum possible measures.
𝑎 = 210 𝑚𝑚
Model by Paul Jackson
1
𝑙 = 𝑏 = 99 𝑚𝑚
3
The construction of the six modules
Francesco Decio’s diagrams
Assembly of six modules
2. The back wall
3. The lateral wall
The cube as a room
1. The floor
And now we proceed by symmetry
Comparison: cube and tetrahedron.
An example of a discussion in the class.
“Now, let's compare the tetrahedron
and the cube that we made ...”
• How many faces do they have? How many vertices? How many edges? Are they regular polyhedra?
• Which is the taller?
• Which among the two has the longest edge?
• Which among the two has the greater volume?
• The cube that has volume……around 53 𝑐𝑚5 = 125 𝑐𝑚3
• What volume will the tetrahedron have?
•
•
•
•
Half the volume of the cube?
Less than half?
More than half?
Almost equal?
Comparison: cube and tetrahedron.
A special question…..
…… the tetrahedron is taller
….. the volume of the tetrahedron is smaller than the volume of the cube
Can you include it in the cube?
Try to believe ………….
The tetrahedron inscribed in the cube
Problem : faced by Kepler in
Epitomes Astronomiae
𝐸𝑑𝑔𝑒𝑇𝑒𝑡𝑟𝑎ℎ𝑒𝑑𝑟𝑜𝑛 =
2 ∙ 𝐸𝑑𝑔𝑒𝐶𝑢𝑏𝑒
The edge of the included tetrahedron is equal to the diagonal of the cube face
Other questions for class discussion
About tetrahedron: what is the distance between two opposite edges?
The distance between two skew lines
………….. is equal to the edge of the cube
How many empty spaces are between the surface of the cube and the included tetrahedron?
What shape do they have? Are they all the same?
They are four pyramids with triangular faces having as base the face of the tetrahedron and
as lateral edges three edges of the cube.
From now we will call these four pyramids, corner pyramids
Now we try to complete the filling of the cube with the four corner
pyramids
𝑎 = 210 𝑚𝑚
We realize the four corner pyramids starting from four
square sheets with a dimension of a third of the short
side of an A4 sheet : square side = 70 mm
1
𝑙 = 𝑎 = 70 𝑚𝑚
3
The costruction of corner pyramids
Inclusion of the tetrahedron and the four corner pyramids in the cube
Consider a question previously placed
The cube is larger than the tetrahedron. How much does it is bigger?
First answer
We know that the cube volume is equal to the included tetrahedron volume
plus the volume of four pyramids added.
All right, but …
.. if we divide the tetrahedron into two pyramids we will obtain six
pyramids in the cube... and this help us…
How can we dissect the tetrahedron?
Dissection of the tetrahedron into two equal pyramids
How to do?
Tetrahedron
Tetrahedron dissected
The pyramids sections of the tetrahedron
By cutting the tetrahedron with a plane through an edge and midpoint of opposite edge you
get two identical pyramids with the base as the same as the tetrahedron and side edges as
the same as the cube edges.
Which part of the cube is the included tetrahedron?
The two pyramids “half tetrahedron" and the four corner pyramids have the
same bases, in fact their bases are the faces of the tetrahedron.
• The six pyramids have the same bases, and what about the heights?
• The heights of six pyramids are equal and then ... ....
• The six pyramids are equivalent so they have the same volume, each is
one-sixth of the volume of the cube.
The volume of the tetrahedron included in cube is
?
?
of the volume of the cube…
What is the volume of the include tetrahedron related to the
volume of the cube?
𝟏
𝟑
The volume of the tetrahedron is of the volume of the cube in which it is inscribed
𝑉𝑜𝑙𝑢𝑚𝑒𝑇𝑒𝑡𝑟𝑎ℎ𝑒𝑑𝑟𝑜𝑛
1
= 𝑉𝑜𝑙𝑢𝑚𝑒𝐶𝑢𝑏𝑒
3
Relations between the paper format and the edge of the cube
We place the short side of the A4 sheet equal to 1: 𝑎 = 1
The edge of the cube is half of the side of square sheet
obtained by dividing into three parts the long edge of
the paper A4.
𝑆𝐶𝑢𝑏𝑒
1 1
2
= ∙ ∙ 2=
2 3
6
2
The edge of the tetrahedron is 3 from the
middle of the short side of the paper A4:
𝑆𝑇𝑒𝑡𝑟𝑎ℎ𝑒𝑑𝑟𝑜𝑛 =
2 1
∙
3 2
𝑆𝑇𝑒𝑡𝑟𝑎ℎ𝑒𝑑𝑟𝑜𝑛
= 2
𝑆𝐶𝑢𝑏𝑒
=
1
3
Relationship between the sheet size and the edges of the angular pyramids
The angular pyramids have:
as the base edges the edge of tetrahedron;
as the lateral edges the edges of the cube.
Then…
𝑠𝑏𝑎𝑠𝑒
1
=
3
𝑠𝑙𝑎𝑡𝑒𝑟𝑎𝑙 𝑒𝑑𝑔𝑒𝑠 =
A third short side A4
1 1
2
∙ ∙ 2=
2 3
6
A sixth long side A4
Calculation of the volume of the tetrahedron of edge 2
The height of the tetrahedron is a cathetus of the right
triangle which has as its hypotenuse the apotema and as
other cathetus the radius of the circle inscribed in the
equilateral triangle base of the tetrahedron.
The volume of the tetrahedron inscribed in that cube is one third of that of the cube
Thank you!
[email protected]
Paper
Antonio Criscuolo, Francesco Decio, Inclusione di un tetraedro regolare in un
cubo a partire da fogli A4, Quadrato Magico CDO n.111 2014-2015